The quadratic assignment problem with a monotone ... - Springer Link
Mathematical Programming 82 (1998) 125 158
The quadratic assignment problem with a monotone anti-Monge and a symmetric Toeplitz matrix: Easy and hard cases Rainer E. Burkard, Eranda Cela *, Giinter Rote, Gerhard J. Woeginger Technische Universit~t Graz, Institut j'ur Mathematik B, Steyrergasse 30, A-8010 Graz, Austria Received 3 July 1995; accepted 2 February 1996
Abstract
This paper investigates a restricted version of the Quadratic Assignment Problem (QAP), where one of the coefficient matrices is an Anti-Monge matrix with non-decreasing rows and columns and the other coefficient matrix is a symmetric Toeplitz matrix. This restricted version is called the Anti-Monge Toeplitz QAP. There are three well-known combinatorial problems that can be modeled via the Anti-Monge Toeplitz QAP: (P1) The "Turbine Problem", i.e. the assignment of given masses to the vertices of a regular polygon such that the distance of the center of gravity of the resulting system to the center of the polygon is minimized. (P2) The Traveling Salesman Problem on symmetric Monge distance matrices. (P3) The arrangement of data records with given access probabilities in a linear storage medium in order to minimize the average access time. We identify conditions on the Toeplitz matrix B that lead to a simple solution for the Anti-Monge-Toeplitz QAP: The optimal permutation can be given in advance without regarding the numerical values of the data. The resulting theorems generalize and unify several known results on problems (P1), (P2), and (P3). We also show that the Turbine Problem is NP-hard and consequently, that the Anti-Monge Toeplitz QAP is NPhard in general. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V. Keywords." Quadratic assignment; Special cases; Polynomially solvable; Anti-Monge matrices; Toeplitz matrices
Dedicated to the m e m o r y oJ" Gene L a w l e r
*Corresponding author. E-mail: [email protected]. 1This research has been supported by the Spezialforschungsbereich F 003 "Optimierung und Kontrolle", Projektbereich Diskrete Optimierung. 0025-5610/98/$19.00 © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V. PHS0025-56 10(97)00085-3
R.E. Burkard et al. / Mathematical Programming82 (1998) 125 158
126 1. Introduction
Given two n × n matrices A = (a~j) and B = (b/j), 1 ~ a i s + a r j for all l ~ < i < r ~ < n and l ~ < j < s ~ < n (this inequality is called the Anti-Monge inequality). Matrix A is called monotone if a~i 3. Notice that, if all diagonal entries of matrix B are equal to a certain constant, the value of this constant does not influence the optimal solution of QAP(A, B) but only its optimal value, for any arbitrary matrix A. Thus, as the QAP we are investigating is the Anti-Monge-Toeplitz QAP, we may assume that all Toeplitz matrices in this section have 0-entries on the diagonal. Consequently, the equality f ( 0 ) = 0 holds for any function f generating some Toeplitz matrix in this section. The proof of the theorem is quite involved although it is based on a simple idea. Before going into technicalities we describe the underlying idea.
2.1. A sketch of the proof We want to prove that 7c* solves QAP(A, B) when A is a non-negative monotone Anti-Monge matrix and B is a Toeplitz matrix generated by a benevolent function f with f ( 0 ) = 0. We will actually prove an even stronger statement. Consider the relaxation of QAP(A, B) where the columns and the rows of matrix A may be permuted independently of each other, the columns according to some permutation ~ and the rows according to some permutation qS. min ~
ag,(O~(i)b~ j.
(1)
¢,~cSn i=l j = l
We call this problem independent-QAP(A,B), and we denote its objective function by n
i 1 j=l
R.E. Burkard et aL / Mathematical Programming 82 (1998) 125-158
132
For q5 = 0 we have the objective function of the usual QAP, and we will then also use the simplified notation
Z((~,A,B) := Z(~, O,A,B). Even for the relaxed problem, the independent-QAP(A, B), it will turn out that the double sum in (1) is minimized by ~b = 0 ~- ~*. This trivially guarantees that ~* is also an optimal solution of QAP(A, B). Thus, Theorem 1.9 can be derived as a corollary of the following theorem. Theorem 2.1. The pair of permutations (~*, 7c*)solves the independent-QAP(A, B) when A is a non-negative monotone Anti-Monge matrix andB is a symmetric Toeplitz matrix which is generated by a non-negative benevolent function f with f(O) = O. In the proof of the theorem, we will see that the set ,~' of non-negative monotone Anti-Monge matrices is a cone, and likewise, the set ~ o f non-negative benevolent matrices with zeros on the diagonal is a cone. It is sufficient to prove the theorem for the extreme rays of these cones, as we shall demonstrate shortly. Since these extreme rays have a simple structure, the proof goes through. In Appendix A it is shown that the extreme rays of the cone s¢ are generated by ~ (Pq)' -+(Pq)= 1 for n - p + 1 ~~2m - 2.
R.E. Burkard et al. / Mathematical Programming 82 (1998) 125-158
137
Proof. Only the last expression is n o t obvious, b u t it can be checked by elementary calculations, which we omit. [] T h r o u g h the rest of this section B is a Toeplitz m a t r i x generated by the f u n c t i o n h ~ for some 1 ~ 7 the m a x i m u m value o f xi is ? and it occurs Jbr q - 7 + 1 adjacent positions (rows). (U2) I f q ~7 a n d p > ~ n - ( q - 7 + l ) ,
(p+7-n)q
if
q ~ n - ( y - q + l ) ,
L(~)eJ
if
b-q-,l÷l
The quadratic assignment problem with a monotone anti-Monge and a symmetric Toeplitz matrix: Easy and hard cases Rainer E. Burkard, Eranda Cela *, Giinter Rote, Gerhard J. Woeginger Technische Universit~t Graz, Institut j'ur Mathematik B, Steyrergasse 30, A-8010 Graz, Austria Received 3 July 1995; accepted 2 February 1996
Abstract
This paper investigates a restricted version of the Quadratic Assignment Problem (QAP), where one of the coefficient matrices is an Anti-Monge matrix with non-decreasing rows and columns and the other coefficient matrix is a symmetric Toeplitz matrix. This restricted version is called the Anti-Monge Toeplitz QAP. There are three well-known combinatorial problems that can be modeled via the Anti-Monge Toeplitz QAP: (P1) The "Turbine Problem", i.e. the assignment of given masses to the vertices of a regular polygon such that the distance of the center of gravity of the resulting system to the center of the polygon is minimized. (P2) The Traveling Salesman Problem on symmetric Monge distance matrices. (P3) The arrangement of data records with given access probabilities in a linear storage medium in order to minimize the average access time. We identify conditions on the Toeplitz matrix B that lead to a simple solution for the Anti-Monge-Toeplitz QAP: The optimal permutation can be given in advance without regarding the numerical values of the data. The resulting theorems generalize and unify several known results on problems (P1), (P2), and (P3). We also show that the Turbine Problem is NP-hard and consequently, that the Anti-Monge Toeplitz QAP is NPhard in general. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V. Keywords." Quadratic assignment; Special cases; Polynomially solvable; Anti-Monge matrices; Toeplitz matrices
Dedicated to the m e m o r y oJ" Gene L a w l e r
*Corresponding author. E-mail: [email protected]. 1This research has been supported by the Spezialforschungsbereich F 003 "Optimierung und Kontrolle", Projektbereich Diskrete Optimierung. 0025-5610/98/$19.00 © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V. PHS0025-56 10(97)00085-3
R.E. Burkard et al. / Mathematical Programming82 (1998) 125 158
126 1. Introduction
Given two n × n matrices A = (a~j) and B = (b/j), 1 ~ a i s + a r j for all l ~ < i < r ~ < n and l ~ < j < s ~ < n (this inequality is called the Anti-Monge inequality). Matrix A is called monotone if a~i 3. Notice that, if all diagonal entries of matrix B are equal to a certain constant, the value of this constant does not influence the optimal solution of QAP(A, B) but only its optimal value, for any arbitrary matrix A. Thus, as the QAP we are investigating is the Anti-Monge-Toeplitz QAP, we may assume that all Toeplitz matrices in this section have 0-entries on the diagonal. Consequently, the equality f ( 0 ) = 0 holds for any function f generating some Toeplitz matrix in this section. The proof of the theorem is quite involved although it is based on a simple idea. Before going into technicalities we describe the underlying idea.
2.1. A sketch of the proof We want to prove that 7c* solves QAP(A, B) when A is a non-negative monotone Anti-Monge matrix and B is a Toeplitz matrix generated by a benevolent function f with f ( 0 ) = 0. We will actually prove an even stronger statement. Consider the relaxation of QAP(A, B) where the columns and the rows of matrix A may be permuted independently of each other, the columns according to some permutation ~ and the rows according to some permutation qS. min ~
ag,(O~(i)b~ j.
(1)
¢,~cSn i=l j = l
We call this problem independent-QAP(A,B), and we denote its objective function by n
i 1 j=l
R.E. Burkard et aL / Mathematical Programming 82 (1998) 125-158
132
For q5 = 0 we have the objective function of the usual QAP, and we will then also use the simplified notation
Z((~,A,B) := Z(~, O,A,B). Even for the relaxed problem, the independent-QAP(A, B), it will turn out that the double sum in (1) is minimized by ~b = 0 ~- ~*. This trivially guarantees that ~* is also an optimal solution of QAP(A, B). Thus, Theorem 1.9 can be derived as a corollary of the following theorem. Theorem 2.1. The pair of permutations (~*, 7c*)solves the independent-QAP(A, B) when A is a non-negative monotone Anti-Monge matrix andB is a symmetric Toeplitz matrix which is generated by a non-negative benevolent function f with f(O) = O. In the proof of the theorem, we will see that the set ,~' of non-negative monotone Anti-Monge matrices is a cone, and likewise, the set ~ o f non-negative benevolent matrices with zeros on the diagonal is a cone. It is sufficient to prove the theorem for the extreme rays of these cones, as we shall demonstrate shortly. Since these extreme rays have a simple structure, the proof goes through. In Appendix A it is shown that the extreme rays of the cone s¢ are generated by ~ (Pq)' -+(Pq)= 1 for n - p + 1 ~~2m - 2.
R.E. Burkard et al. / Mathematical Programming 82 (1998) 125-158
137
Proof. Only the last expression is n o t obvious, b u t it can be checked by elementary calculations, which we omit. [] T h r o u g h the rest of this section B is a Toeplitz m a t r i x generated by the f u n c t i o n h ~ for some 1 ~ 7 the m a x i m u m value o f xi is ? and it occurs Jbr q - 7 + 1 adjacent positions (rows). (U2) I f q ~7 a n d p > ~ n - ( q - 7 + l ) ,
(p+7-n)q
if
q ~ n - ( y - q + l ) ,
L(~)eJ
if
b-q-,l÷l
